Autoregressive time series analysis via representatives

Abstract

In any realization of an autoregressive time series there exist a few observations having a noticeable feature: they express the useful properties of the time series and, therefore, they represent the entire process. Such representative observations (or, simply, representatives) can be determined by an optimization procedure, provided that the absolute value criterion is used instead of the customary least squares. To achieve this, a special kind of optimization operator (optimator) which generate the parameters of the time series is considered. The concepts of strong and weak similarity of the time series are defined in terms of the representatives and sufficient conditions for both strong and weak similarity are derived. It is shown that there exists a subclass of strongly similar processes, say X such that ordinary addition is a binary operation in X. An analogous result is shown to hold for weakly similar autoregressive processes. Some examples illustrating these results are given.